Strategy Variations and Algebra Readiness: Examining Procedural Flexibility Using Latent Profile Analysis

• In Event: The Importance of Mathematical Competencies: When and for Whom Do They Matter?

Abstract

Purpose
The purpose of this study is to examine variations in middle-schoolers’ demonstrated procedural flexibility when solving fraction arithmetic problems and how these variations relate to end-of-year algebra readiness.
Theoretical Framework
Procedural flexibility is a skill crucial to developing mathematical knowledge (Durkin et al,. 2017; 2021; Star & Newton, 2009). When students demonstrate procedural flexibility, they apply a situationally appropriate strategy that is in some way better than the standard method of problem-solving (i.e., the standard algorithm; Newton et al., 2020; Star, 2005; Star et al., 2022). However, which strategies are considered optimal vary depending on the learner, task, and environment in which problem-solving occurs (Star et al., 2022; Verschaffel et al., 2007). Further, variations in strategy use may be differentially related to algebra skills because each strategy requires different mathematical competencies to execute correctly.
Methods
Students in Grades 6-8 (N = 350) taking part in a larger study completed paper-and-pencil measures at the start (Fall 2017) and end (Spring 2018) of the school year.
We used latent profile analysis (LPA) to examine individual patterns in students’ strategy use when solving fraction arithmetic problems to illuminate what strategies they use. We also used multinomial logistic regression to examine predictors of most likely profile membership and the BCH method to examine whether strategy profiles predict end of year algebra achievement.
Materials and Coding Procedure
We measured students’ performance on fraction arithmetic problems (e.g., 2 ½ × 4 = __), algebraic feature knowledge (conceptual understanding of algebraic concepts, e.g., variables), algebra equation-solving (procedural knowledge of problem-solving), and fraction magnitude knowledge (fraction number line estimation).
We coded fraction arithmetic problems for various types of strategy use: direct, indirect, cancellation, answer only, and standard algorithm. See Table 1 for definitions and examples. We used the proportion of times students exhibited each strategy as LPA indicators.
Results
Students used a variety of strategies aside from the standard algorithm when solving fraction arithmetic problems. Students’ strategy-use grouped into three profiles: Prototypical strategy use (i.e., primarily using the standard algorithm), Atypical strategy use (i.e., primarily using cancellation), and Distributed strategy use (i.e., relatively even use of each strategy type; Figure 1 and Table 1). Grade level and algebra equation-solving skills predicted profile membership (Table 2). Students who used an indirect problem-solving strategy were more likely to display the Distributed profile than the other two profiles. Students who were most likely to display the Distributed profile and the Atypical profile showed higher end-of-year algebra readiness (Table 3).
Significance
We found that using a variety of flexible problem-solving strategies and displaying a Distributed strategy profile is associated with critical algebra skills, including algebraic feature knowledge and equation-solving. These findings provide evidence that flexible strategy use that is accurate but not necessarily efficient can still be related to beneficial mathematical outcomes and lay groundwork to expand upon what strategies are considered situationally appropriate.

Authors

• Elena Silla, University of Delaware

• Kristie Newton, Temple University

• Christina Areizaga Barbieri, University of Delaware